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On Lucas pseudoprimes of the form \(ax^ 2+bxy+cy^ 2\). (English) Zbl 0852.11006
Bergum, G. E. (ed.) et al., Applications of Fibonacci numbers. Volume 6: Proceedings of the sixth international research conference on Fibonacci numbers and their applications, Washington State University, Pullman, WA, USA, July 18-22, 1994. Dordrecht: Kluwer Academic Publishers. 409-421 (1996).
Lucas numbers \(\{L_k\}\) are defined by the formula \(L_k= L\cdot L_{k-1}- M\cdot L_{k-2}\) for \(k\geq 2\), and \(L_0= 0\), \(L_1 =1\), where \(L\), \(M\) are rational integers, \(L>0\). A Lucas pseudoprime is an odd composite integer \(n\) coprime to \(D= L^2- 4M\) and satisfying \(L_{n- (D/n)} \equiv 0\pmod n\), where \((D/n)\) is the Jacobi symbol. Generalizing an earlier result [cf. the author and A. Schinzel, C. R. Acad. Sci., Paris 258, 3617-3620 (1964; Zbl 0117.28203)] the author proves that under some mild restrictions every primitive binary quadratic form \(ax^2+ bxy+ cz^2\) represents infinitely many Lucas pseudoprimes.
For the entire collection see [Zbl 0836.00026].

MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11E16 General binary quadratic forms
11A15 Power residues, reciprocity
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